The radius of gyration of a cylindrical rod about an axis of rotation perpendicular to its length and passing through its center is $k$. Given that the length of the rod is $10 \sqrt{3} \ m$,find the value of $k$ in meters.

What is the radius of gyration? Write its unit and dimensional formula.

Two discs of the same material and thickness have radii $0.2\, m$ and $0.6\, m.$ Their moments of inertia about their axes will be in the ratio

$A$ uniform solid cylinder with radius $R$ and length $L$ has a moment of inertia $I_1$ about its central axis. $A$ concentric solid cylinder of radius $R' = \frac{R}{2}$ and length $L' = \frac{L}{2}$ is carved out of the original cylinder. If $I_2$ is the moment of inertia of the carved-out portion of the cylinder about the same axis,then $\frac{I_1}{I_2} = ..........$

The moment of inertia of a uniform thin rod of length $L$ and mass $M$ about an axis passing through a point at a distance of $\frac{L}{3}$ from one of its ends and perpendicular to the rod is

The moment of inertia of a uniform rod of mass $M$ and length $L$ about an axis passing through its center and perpendicular to it is $\frac{1}{12} ML^2$. The rod is bent at the center such that the two halves make an angle of $60^\circ$ with each other. Find the moment of inertia of the bent rod about the same axis passing through the center of the rod (the point of bending) and perpendicular to the plane containing the two halves.